I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.

Do you know where I can find a complete proof of such a result : There exists a unique solution $u$ to the problem

$\left\{ \begin{array}{rcll} \partial_t u(t,x) - \eta \Delta u(t,x) &=& f(t,x)&\qquad\text{on } ]0,T]\times \mathbb{R}^n \\ u(0,x) &=& u_0(x)&\qquad\text{on } \mathbb{R}^n \end{array} \right.$

with $f\in L^2(0,T;L^2(\mathbb{R}^n))$, $u_0 \in H^1(\mathbb{R}^n)$ and $\eta>0$ fixed and furthermore $u\in L^2(0,T;H^2(\mathbb{R}^n))$ and $\partial_t u \in L^2(0,T;L^2(\mathbb{R}^n))$.

Thank you